Modern pure mathematics suffers from a uniform disinterest in examining the foundations of the subject carefully and objectively. The current belief system that "mathematics is based on set theory" is quite misguided, and in its current form represents an abdication of our responsibility to the integrity of our discipline. We have essentially outsourced the very heart of our subject to philosophy.

In this video we set out our campaign to put modern "Set Theory", in the sense of "hierarchies upon hierarchies of infinite sets" on trial. And we give an overview of how and why we got into the current sad state where everyone believes and dissent is considered scandalous.

Cantor and Dedekind and then Hilbert are perhaps mostly responsible for the big changes pushed, against the will of most of their contemporaries, into modern pure mathematics, culminating in the axiomatic frameworks introduced by logicians in the first half of the 20th century.

Mathematics needs to find a more sensible, and logical starting positionnot only to make the subject cleaner and more exciting for students and researchers, but also to make room for the new forces about to be unleashed into pure mathematics: the next generation of AI machines.

Reaction to some comments: Some people are suggesting that I rarely give details to my objections. To those who have watched a lot of my videos, this is clearly untrue: I have very clearly set out many different objections for example to real numbers earlier in this MathFoundations series. In particular, my concrete question of: what exactly is the sum pi+e+sqrt(2) ? never seems to get answered.

Video Content:
00:00 Does modern set theory really work as a logical foundation?
01:40 Modern set theory
03:41 Arithmetic with natural numbers as the mathematical foundation
05:11 How to model the continuum in mathematics
08:51 Ancient Greeks, 17th and 18th century, analysis
09:33 19th century mathematical analysis
13:11 20th century mathematical analysis

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My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/...

My blog is at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things.

Online courses are being developed at openlearning.com. The first one, already underway, is Algebraic Calculus One at https://www.openlearning.com/courses/... Please join us for an exciting new approach to one of mathematics' most important subjects!

If you would like to support these new initiatives for mathematics education and research, please consider becoming a Patron of this Channel at   / njwildberger   Your support would be much appreciated.

Here are the Insights into Mathematics Playlists:

   • MathHistory: A course in the History ...  
   • WildTrig: Intro to Rational Trigonometry  
   • Math Foundations  
   • Wild Linear Algebra  
   • Famous Math Problems  
   • Box Arithmetic  
   • Elementary Mathematics (K6) Explained  
   • Year9Maths  
   • Ancient Mathematics  
   • Wild West Banking  
   • Sociology and Pure Mathematics  
   • Sociology and Pure Physics  
   • Old Babylonian Mathematics  
   • Probability and Statistics: an introd...  
   • Boole's Logic and Circuit Analysis  
   • Universal Hyperbolic Geometry  
   • Differential Geometry  
   • Algebraic Topology  
   • MathSeminars  
   • Playing Go  
   • Diffusion Symmetry: A bridge between ...